refactor(linear_algebra/charpoly): split file to reduce imports (#13778)

While working on representation theory I was annoyed to find that essentially all of field theory was being transitively imported (causing lots of unnecessary recompilation). This improves the situation slightly.



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

Author: kim-em

archive/100-theorems-list/83_friendship_graphs.lean

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark, Kyle Miller
 -/
 import combinatorics.simple_graph.adj_matrix
-import linear_algebra.matrix.charpoly.coeff
+import linear_algebra.matrix.charpoly.finite_field
 import data.int.modeq
 import data.zmod.basic
 import tactic.interval_cases

src/linear_algebra/charpoly/basic.lean

@@ -6,6 +6,7 @@ Authors: Riccardo Brasca
 
 import linear_algebra.free_module.finite.basic
 import linear_algebra.matrix.charpoly.coeff
+import field_theory.minpoly
 
 /-!
 
@@ -30,6 +31,7 @@ open_locale classical matrix polynomial
 
 noncomputable theory
 
+
 open module.free polynomial matrix
 
 namespace linear_map

src/linear_algebra/matrix/charpoly/coeff.lean

@@ -4,15 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Jalex Stark
 -/
 
-import algebra.polynomial.big_operators
-import data.matrix.char_p
-import field_theory.finite.basic
-import group_theory.perm.cycles
+import data.polynomial.expand
 import linear_algebra.matrix.charpoly.basic
-import linear_algebra.matrix.trace
-import linear_algebra.matrix.to_lin
-import ring_theory.polynomial.basic
-import ring_theory.power_basis
 
 /-!
 # Characteristic polynomials
@@ -182,54 +175,8 @@ begin
       not_false_iff] }
 end
 
-@[simp] lemma finite_field.matrix.charpoly_pow_card {K : Type*} [field K] [fintype K]
-  (M : matrix n n K) : (M ^ (fintype.card K)).charpoly = M.charpoly :=
-begin
-  casesI (is_empty_or_nonempty n).symm,
-  { cases char_p.exists K with p hp, letI := hp,
-    rcases finite_field.card K p with ⟨⟨k, kpos⟩, ⟨hp, hk⟩⟩,
-    haveI : fact p.prime := ⟨hp⟩,
-    dsimp at hk, rw hk at *,
-    apply (frobenius_inj K[X] p).iterate k,
-    repeat { rw iterate_frobenius, rw ← hk },
-    rw ← finite_field.expand_card,
-    unfold charpoly, rw [alg_hom.map_det, ← coe_det_monoid_hom,
-      ← (det_monoid_hom : matrix n n K[X] →* K[X]).map_pow],
-    apply congr_arg det,
-    refine mat_poly_equiv.injective _,
-    rw [alg_equiv.map_pow, mat_poly_equiv_charmatrix, hk, sub_pow_char_pow_of_commute, ← C_pow],
-    { exact (id (mat_poly_equiv_eq_X_pow_sub_C (p ^ k) M) : _) },
-    { exact (C M).commute_X } },
-  { -- TODO[gh-6025]: remove this `haveI` once `subsingleton_of_empty_right` is a global instance
-    haveI : subsingleton (matrix n n K) := subsingleton_of_empty_right,
-    exact congr_arg _ (subsingleton.elim _ _), },
-end
-
-@[simp] lemma zmod.charpoly_pow_card (M : matrix n n (zmod p)) :
-  (M ^ p).charpoly = M.charpoly :=
-by { have h := finite_field.matrix.charpoly_pow_card M, rwa zmod.card at h, }
-
-lemma finite_field.trace_pow_card {K : Type*} [field K] [fintype K]
-  (M : matrix n n K) : trace (M ^ (fintype.card K)) = trace M ^ (fintype.card K) :=
-begin
-  casesI is_empty_or_nonempty n,
-  { simp [zero_pow fintype.card_pos, matrix.trace], },
-  rw [matrix.trace_eq_neg_charpoly_coeff, matrix.trace_eq_neg_charpoly_coeff,
-       finite_field.matrix.charpoly_pow_card, finite_field.pow_card]
-end
-
-lemma zmod.trace_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
-  trace (M ^ p) = (trace M)^p :=
-by { have h := finite_field.trace_pow_card M, rwa zmod.card at h, }
-
 namespace matrix
 
-theorem is_integral : is_integral R M := ⟨M.charpoly, ⟨charpoly_monic M, aeval_self_charpoly M⟩⟩
-
-theorem minpoly_dvd_charpoly {K : Type*} [field K] (M : matrix n n K) :
-  (minpoly K M) ∣ M.charpoly :=
-minpoly.dvd _ _ (aeval_self_charpoly M)
-
 /-- Any matrix polynomial `p` is equivalent under evaluation to `p %ₘ M.charpoly`; that is, `p`
 is equivalent to a polynomial with degree less than the dimension of the matrix. -/
 lemma aeval_eq_aeval_mod_charpoly (M : matrix n n R) (p : R[X]) :
@@ -243,29 +190,3 @@ lemma pow_eq_aeval_mod_charpoly (M : matrix n n R) (k : ℕ) : M^k = aeval M (X^
 by rw [←aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
 
 end matrix
-
-section power_basis
-
-open algebra
-
-/-- The characteristic polynomial of the map `λ x, a * x` is the minimal polynomial of `a`.
-
-In combination with `det_eq_sign_charpoly_coeff` or `trace_eq_neg_charpoly_coeff`
-and a bit of rewriting, this will allow us to conclude the
-field norm resp. trace of `x` is the product resp. sum of `x`'s conjugates.
--/
-lemma charpoly_left_mul_matrix {K S : Type*} [field K] [comm_ring S] [algebra K S]
-  (h : power_basis K S) :
-  (left_mul_matrix h.basis h.gen).charpoly = minpoly K h.gen :=
-begin
-  apply minpoly.unique,
-  { apply matrix.charpoly_monic },
-  { apply (injective_iff_map_eq_zero (left_mul_matrix _)).mp (left_mul_matrix_injective h.basis),
-    rw [← polynomial.aeval_alg_hom_apply, aeval_self_charpoly] },
-  { intros q q_monic root_q,
-    rw [matrix.charpoly_degree_eq_dim, fintype.card_fin, degree_eq_nat_degree q_monic.ne_zero],
-    apply with_bot.some_le_some.mpr,
-    exact h.dim_le_nat_degree_of_root q_monic.ne_zero root_q }
-end
-
-end power_basis

src/linear_algebra/matrix/charpoly/finite_field.lean

@@ -0,0 +1,60 @@
+/-
+Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Aaron Anderson, Jalex Stark
+-/
+import linear_algebra.matrix.charpoly.coeff
+import field_theory.finite.basic
+import data.matrix.char_p
+
+
+/-!
+# Results on characteristic polynomials and traces over finite fields.
+-/
+
+noncomputable theory
+
+open polynomial matrix
+open_locale polynomial
+
+variables {n : Type*} [decidable_eq n] [fintype n]
+
+@[simp] lemma finite_field.matrix.charpoly_pow_card {K : Type*} [field K] [fintype K]
+  (M : matrix n n K) : (M ^ (fintype.card K)).charpoly = M.charpoly :=
+begin
+  casesI (is_empty_or_nonempty n).symm,
+  { cases char_p.exists K with p hp, letI := hp,
+    rcases finite_field.card K p with ⟨⟨k, kpos⟩, ⟨hp, hk⟩⟩,
+    haveI : fact p.prime := ⟨hp⟩,
+    dsimp at hk, rw hk at *,
+    apply (frobenius_inj K[X] p).iterate k,
+    repeat { rw iterate_frobenius, rw ← hk },
+    rw ← finite_field.expand_card,
+    unfold charpoly, rw [alg_hom.map_det, ← coe_det_monoid_hom,
+      ← (det_monoid_hom : matrix n n K[X] →* K[X]).map_pow],
+    apply congr_arg det,
+    refine mat_poly_equiv.injective _,
+    rw [alg_equiv.map_pow, mat_poly_equiv_charmatrix, hk, sub_pow_char_pow_of_commute, ← C_pow],
+    { exact (id (mat_poly_equiv_eq_X_pow_sub_C (p ^ k) M) : _) },
+    { exact (C M).commute_X } },
+  { -- TODO[gh-6025]: remove this `haveI` once `subsingleton_of_empty_right` is a global instance
+    haveI : subsingleton (matrix n n K) := subsingleton_of_empty_right,
+    exact congr_arg _ (subsingleton.elim _ _), },
+end
+
+@[simp] lemma zmod.charpoly_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
+  (M ^ p).charpoly = M.charpoly :=
+by { have h := finite_field.matrix.charpoly_pow_card M, rwa zmod.card at h, }
+
+lemma finite_field.trace_pow_card {K : Type*} [field K] [fintype K]
+  (M : matrix n n K) : trace (M ^ (fintype.card K)) = trace M ^ (fintype.card K) :=
+begin
+  casesI is_empty_or_nonempty n,
+  { simp [zero_pow fintype.card_pos, matrix.trace], },
+  rw [matrix.trace_eq_neg_charpoly_coeff, matrix.trace_eq_neg_charpoly_coeff,
+       finite_field.matrix.charpoly_pow_card, finite_field.pow_card]
+end
+
+lemma zmod.trace_pow_card {p : ℕ} [fact p.prime] (M : matrix n n (zmod p)) :
+  trace (M ^ p) = (trace M)^p :=
+by { have h := finite_field.trace_pow_card M, rwa zmod.card at h, }

src/linear_algebra/matrix/charpoly/minpoly.lean

@@ -0,0 +1,61 @@
+/-
+Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Aaron Anderson, Jalex Stark
+-/
+
+import linear_algebra.matrix.charpoly.coeff
+import ring_theory.power_basis
+
+/-!
+# The minimal polynomial divides the characteristic polynomial of a matrix.
+-/
+
+noncomputable theory
+
+universes u v
+
+open polynomial matrix
+
+variables {R : Type u} [comm_ring R]
+variables {n : Type v} [decidable_eq n] [fintype n]
+
+open finset
+
+variable {M : matrix n n R}
+
+namespace matrix
+
+theorem is_integral : is_integral R M := ⟨M.charpoly, ⟨charpoly_monic M, aeval_self_charpoly M⟩⟩
+
+theorem minpoly_dvd_charpoly {K : Type*} [field K] (M : matrix n n K) :
+  (minpoly K M) ∣ M.charpoly :=
+minpoly.dvd _ _ (aeval_self_charpoly M)
+
+end matrix
+
+section power_basis
+
+open algebra
+
+/-- The characteristic polynomial of the map `λ x, a * x` is the minimal polynomial of `a`.
+
+In combination with `det_eq_sign_charpoly_coeff` or `trace_eq_neg_charpoly_coeff`
+and a bit of rewriting, this will allow us to conclude the
+field norm resp. trace of `x` is the product resp. sum of `x`'s conjugates.
+-/
+lemma charpoly_left_mul_matrix {K S : Type*} [field K] [comm_ring S] [algebra K S]
+  (h : power_basis K S) :
+  (left_mul_matrix h.basis h.gen).charpoly = minpoly K h.gen :=
+begin
+  apply minpoly.unique,
+  { apply matrix.charpoly_monic },
+  { apply (injective_iff_map_eq_zero (left_mul_matrix _)).mp (left_mul_matrix_injective h.basis),
+    rw [← polynomial.aeval_alg_hom_apply, aeval_self_charpoly] },
+  { intros q q_monic root_q,
+    rw [matrix.charpoly_degree_eq_dim, fintype.card_fin, degree_eq_nat_degree q_monic.ne_zero],
+    apply with_bot.some_le_some.mpr,
+    exact h.dim_le_nat_degree_of_root q_monic.ne_zero root_q }
+end
+
+end power_basis

src/ring_theory/norm.lean

@@ -6,7 +6,7 @@ Authors: Anne Baanen
 
 import field_theory.primitive_element
 import linear_algebra.determinant
-import linear_algebra.matrix.charpoly.coeff
+import linear_algebra.matrix.charpoly.minpoly
 import linear_algebra.matrix.to_linear_equiv
 import field_theory.is_alg_closed.algebraic_closure
 

src/ring_theory/trace.lean

@@ -5,7 +5,7 @@ Authors: Anne Baanen
 -/
 
 import linear_algebra.matrix.bilinear_form
-import linear_algebra.matrix.charpoly.coeff
+import linear_algebra.matrix.charpoly.minpoly
 import linear_algebra.determinant
 import linear_algebra.vandermonde
 import linear_algebra.trace